derivedMS: Derived Parameters of Fitted SECR Model

Description

Compute derived parameters of spatially explicit capture-recapture model.
Density is a derived parameter when a model is fitted by maximizing the conditional likelihood. So also is the effective sampling area (in the sense of Borchers and Efford 2008).

Arguments

object

secr object output from secr.fit, or an
object of class c("secrlist", "list")

sessnum

index of session in object$capthist for which output required

groups

vector of covariate names to define group(s) (see Details)

alpha

alpha level for confidence intervals

se.esa

logical for whether to calculate SE(mean(esa))

se.D

logical for whether to calculate SE(D-hat)

loginterval

logical for whether to base interval on log(D)

distribution

character string for distribution of the number of individuals detected

ncores

integer number of cores available for parallel processing

beta

vector of fitted parameters on transformed (link) scale

real

vector of ‘real’ parameters

noccasions

integer number of sampling occasions (see Details)

...

other arguments (not used)

Details

The derived estimate of density is a Horvitz-Thompson-like estimate:

D-hat
= sum( a_i (theta-hat)^--1)

where
(theta-hat) is the estimate of effective sampling area for animal
i with detection parameter vector θ.

A non-null value of the argument distribution overrides the value
in object$details. The sampling variance of D-hat
from secr.fit by default is spatially unconditional
(distribution = "Poisson"). For sampling variance conditional on the population of the
habitat mask (and therefore dependent on the mask area), specify
distribution = "binomial". The equation for the conditional
variance includes a factor (1 - a/A) that disappears in the
unconditional (Poisson) variance (Borchers and Efford 2007). Thus the
conditional variance is always less than the unconditional variance. The
unconditional variance may in turn be an overestimate or (more likely)
an underestimate if the true spatial variance is non-Poisson.

Derived parameters may be estimated for population subclasses (groups)
defined by the user with the groups argument. Each named factor
in groups should appear in the covariates dataframe of
object$capthist (or each of its components, in the case of a
multi-session dataset).

esa is used by derived to compute individual-specific
effective sampling areas:

a_i = integral p.(X; z_i, theta_i) dX

where
p.(X) is the probability an individual at X is
detected at least once and the z_i are optional
individual covariates. Integration is over the area A of the
habitat mask.

The argument noccasions may be used to vary the number of
sampling occasions; it works only when detection parameters are constant
across individuals and across time.

The effective sampling area ‘esa’ (a(theta-hat))
reported by derived is equal to the harmonic mean of the
a_i (theta-hat) (arithmetic
mean prior to version 1.5). The sampling variance of
a(theta-hat) is estimated by

var(a(theta)) = G-hat^T V-hat G-hat,

where V-hat is the asymptotic estimate of the
variance-covariance matrix of the beta detection parameters
(theta) and G-hat is a numerical estimate
of the gradient of a(theta) with respect to
theta, evaluated at theta-hat.

A 100(1–alpha)% asymptotic confidence interval is reported for
density. By default, this is asymmetric about the estimate because the
variance is computed by backtransforming from the log scale. You may
also choose a symmetric interval (variance obtained on natural scale).

The vector of detection parameters for esa may be specified via
beta or real, with the former taking precedence. If
neither is provided then the fitted values in object$fit$par are
used. Specifying real parameter values bypasses the various
linear predictors. Strictly, the ‘real’ parameters are for a naive
capture (animal not detected previously).

The computation of sampling variances is relatively slow and may be
suppressed with se.esa and se.D as desired.

If ncores > 1 the parallel package is used to create
processes on multiple cores (see Parallel for more).

For computing derived across multiple models in parallel see
par.derived.

Value

Dataframe with one row for each derived parameter (‘esa’, ‘D’) and
columns as below

estimate

estimate of derived parameter

SE.estimate

standard error of the estimate

lcl

lower 100(1--alpha)% confidence limit

ucl

upper 100(1--alpha)% confidence limit

CVn

relative SE of number observed (Poisson or binomial assumption)

CVa

relative SE of effective sampling area

CVD

relative SE of density estimate

For a multi-session or multi-group analysis the value is a list
with one component for each session and group.

The result will also be a list if object is an ‘secrlist’.

Warning

derived() may be applied to detection models fitted by maximizing the full likelihood (CL = FALSE). However, models using g (groups) will not be handled correctly.

Note

Before version 2.1, the output table had columns for ‘varcomp1’ (the variance in D-hat due to variation in n, i.e.,
Huggins' s^2), and ‘varcomp2’ (the variance in D-hat due to uncertainty in estimates of detection parameters).